1. **Problem Statement:** We are given a triangle SRT with angles at vertices R and T expressed as $(3x + 1)^\circ$ and $(2x + 2)^\circ$ respectively. The sides SR and ST are congruent, meaning triangle SRT is isosceles with $\angle R = \angle T$. We need to find the measure of $\angle S$. 2. **Key Formula:** The sum of interior angles in any triangle is always $180^\circ$: $$\angle S + \angle R + \angle T = 180^\circ$$ 3. **Using Isosceles Triangle Property:** Since SR = ST, angles opposite these sides are equal: $$\angle R = \angle T$$ 4. **Set up the equation for equal angles:** $$3x + 1 = 2x + 2$$ 5. **Solve for $x$:** $$3x + 1 = 2x + 2$$ $$3x - 2x = 2 - 1$$ $$x = 1$$ 6. **Find the measures of $\angle R$ and $\angle T$ by substituting $x=1$:** $$\angle R = 3(1) + 1 = 4^\circ$$ $$\angle T = 2(1) + 2 = 4^\circ$$ 7. **Find $\angle S$ using the triangle angle sum:** $$\angle S = 180^\circ - (\angle R + \angle T) = 180^\circ - (4^\circ + 4^\circ) = 172^\circ$$ **Final answer:** $$\boxed{\angle S = 172^\circ}$$